# Search result: Catalogue data in Autumn Semester 2021

Mathematics Master | ||||||

Course Units for Additional Admission Requirements The courses below are only available for MSc students with additional admission requirements. | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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406-2004-AAL | Algebra IIEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 5 credits | 11R | M. Burger | |

Abstract | Galois theory and related topics. The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

Objective | Introduction to fundamentals of field extensions, Galois theory, and related topics. | |||||

Content | The main topic is Galois Theory. Starting point is the problem of solvability of algebraic equations by radicals. Galois theory solves this problem by making a connection between field extensions and group theory. Galois theory will enable us to prove the theorem of Abel-Ruffini, that there are polynomials of degree 5 that are not solvable by radicals, as well as Galois' theorem characterizing those polynomials which are solvable by radicals. | |||||

Literature | Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1, Graduate Studies in Mathematics,Volume 165 American Mathematical Society Galois Theory is the topic treated in Chapter A5. | |||||

Prerequisites / Notice | Algebra I, in Rotman's book this corresponds to the topics treated in the Chapters A3 and A4. | |||||

406-2005-AAL | Algebra I and IIEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 12 credits | 26R | M. Burger, M. Einsiedler | |

Abstract | Introduction and development of some basic algebraic structures - groups, rings, fields including Galois theory, representations of finite groups, algebras. The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

Objective | ||||||

Content | Basic notions and examples of groups; Subgroups, Quotient groups and Homomorphisms, Group actions and applications Basic notions and examples of rings; Ring Homomorphisms, ideals, and quotient rings, rings of fractions Euclidean domains, Principal ideal domains, Unique factorization domains Basic notions and examples of fields; Field extensions, Algebraic extensions, Classical straight edge and compass constructions Fundamentals of Galois theory Representation theory of finite groups and algebras | |||||

Literature | Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1, Graduate Studies in Mathematics,Volume 165 American Mathematical Society | |||||

406-2303-AAL | Complex AnalysisEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | T. H. Willwacher | |

Abstract | Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, conformal mappings, Riemann mapping theorem. | |||||

Objective | ||||||

Literature | L. Ahlfors: "Complex analysis. An introduction to the theory of analytic functions of one complex variable." International Series in Pure and Applied Mathematics. McGraw-Hill Book Co. B. Palka: "An introduction to complex function theory." Undergraduate Texts in Mathematics. Springer-Verlag, 1991. R.Remmert: Theory of Complex Functions.. Springer Verlag E.Hille: Analytic Function Theory. AMS Chelsea Publication | |||||

406-2284-AAL | Measure and IntegrationAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | F. Da Lio | |

Abstract | Introduction to the abstract measure theory and integration, including the following topics: Lebesgue measure and Lebesgue integral, Lp-spaces, convergence theorems, differentiation of measures, product measures (Fubini's theorem), abstract measures, Radon-Nikodym theorem, probabilistic language. | |||||

Objective | Basic acquaintance with the theory of measure and integration, in particular, Lebesgue's measure and integral. | |||||

Literature | 1. Lecture notes by Professor Michael Struwe (Link) 2. L. Evans and R.F. Gariepy "Measure theory and fine properties of functions" 3. Walter Rudin "Real and complex analysis" 4. R. Bartle The elements of Integration and Lebesgue Measure 5. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis. Link | |||||

406-2554-AAL | TopologyAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 6 credits | 13R | P. Feller | |

Abstract | Topological spaces, continuous maps, connectedness, compactness, metric spaces, quotient spaces, homotopy, fundamental group and covering spaces, van Kampen Theorem. | |||||

Objective | ||||||

Literature | James Munkres: Topology | |||||

Prerequisites / Notice | The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material. | |||||

406-2604-AAL | Probability and StatisticsAny other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | E- | 7 credits | 15R | J. Teichmann | |

Abstract | Introduction to probability and statistics with many examples, based on chapters from the books "Probability and Random Processes" by G. Grimmett and D. Stirzaker and "Mathematical Statistics and Data Analysis" by J. Rice. | |||||

Objective | The goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. In addition to a mathematically rigorous treatment, also an intuitive understanding and familiarity with the ideas behind the definitions are emphasized. Measure theory is not used systematically, but it should become clear why and where measure theory is needed. | |||||

Content | Probability: Chapters 1-5 (Probabilities and events, Discrete and continuous random variables, Generating functions) and Sections 7.1-7.5 (Convergence of random variables) from the book "Probability and Random Processes". Most of this material is also covered in Chap. 1-5 of "Mathematical Statistics and Data Analysis", on a slightly easier level. Statistics: Sections 8.1 - 8.5 (Estimation of parameters), 9.1 - 9.4 (Testing Hypotheses), 11.1 - 11.3 (Comparing two samples) from "Mathematical Statistics and Data Analysis". | |||||

Literature | Geoffrey Grimmett and David Stirzaker, Probability and Random Processes. 3rd Edition. Oxford University Press, 2001. John A. Rice, Mathematical Statistics and Data Analysis, 3rd edition. Duxbury Press, 2006. |

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